# Regularizing seismic inverse problems : transdimensional and machine learning based strategies

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Seismic inversion is a well-established technique in geophysics used to generate quantitative estimates of subsurface rock properties, such as lithology, porosity, fluid content and density from seismic data. It is an iterative process where an initial model is updated based on the comparison between the observed data and the synthetic data generated by simulating the propagation of seismic waves using approximations of the wave equation. However, such inverse problems are high-dimensional and highly non-linear. These problems are mathematically ill-posed resulting in non-uniqueness.The non-uniqueness can be attributed to incomplete data coverage, inaccurate forward model and noise in the data. In this thesis, I address the problem of regularizing such inverse problems. Regularization aims to mitigate the effect of noise and incomplete data coverage by introducing addtional constraints in the inverse problems, thus improving the stability of the problem and preventing unrealistic or erratic results. I study both stochastic and deterministic inverse algorithms. Stochastic seismic inversion takes into consideration the ill-posedness and uncertainty by incorporating probabilistic methods into the inversion process. The Bayesian approach to stochastic inversion provides a natural framework for uncertainty quantification. Markov Chain Monte Carlo (MCMC) sampling is the most common Bayesian inference method. Traditional MCMC algorithms when applied to the same problem, presume and fix the model parameterization, which leads to an overfitting problem to the noise in the data. Hence, in order to reduce the overfitting, a subjective regularization is imposed on the problem. Conventional MCMC algorithms widely used for geophysical inverse problems presume and fix the number of model parameters. However, reversible jump MCMC (RJMCMC) allows the number of model parameters (model dimensionality) to vary during the inversion process and thus appropriate model complexity is directly inferred from data and the prior distribution. However, current implementation of the transdimensional RJMCMC algorithms do not take into account the multimodal distribution of elastic properties and honour the rock physics relationship among themelastic properties. To address this problem, I extend the RJMCMC method to the problem of discrete-continuous seismic inverse problem where I simultaneously invert for facies and elastic reservoir properties from pre-stack seismic data. Secondly, I looked into the application of machine learning algorithms for seismic inverse problems. Conventional machine learning algorithms designed directly to map seismic data to desired properties donot take into account physics based constraints or produce robust uncertainty estimates. Moreover, they require a large amount of training data. To overcome this issue, an Invertible Neural Network is designed to estimate elastic and petrophysical properties from seismic data. INN establishes bijective mappings between the input (physical model) and the output (observed data) and introduces an additional latent output variable to capture the information that is otherwise lost during the forward modeling process. The latent variable can be used to estimate the complete posterior distribution of model parameters. Finally, I designed workflows based on Physics Guided Machine Learning paradigm for full waveform inversion. Unlike traditional machine learning algorithms where one trains the network using labelled seismic data-velocity pairs, I use the physics of wave propagation to train the network. My physics guided network overcomes several issues faced by conventional full waveform inversion algorithms like cycle skipping and inter parameter crosstalk. Cycle skipping refers to situations where FWI fails to find the correct solution due to inadequate initial model assumptions, absence of low frequencies or the presence of highly complex subsurface features. The cycle-skipping issue is further exacerbated in case of multiparameter full waveform inversion. Interparameter crosstalk occurs due to coupling effects between different parameters. The coupling effects between different parameters impedes convergence to global minima since the misfit caused by the inaccuracy in the estimate of one model parameter are wrongly ascribed to a different model parameter.